Remark 2 The upper bound in this result (\(2k=n\) ) gives sets that saturate a known bound [1]}, [2]} for the size of one-way distinguishable sets of maximally entangled states on \(\mathbb {C}^d\otimes \mathbb {C}^d\)  (\(d=2^n\) ). For \(2k<n\) , this produces (non-trivial) distinguishable sets, which is significant as it is known [3]} that many sets defined from \(\mathcal {P}_n\)  with less than \(2^n\)  operators (here \(4^k < 2^n\) ) are not distinguishable even with positive partial transpose operations, and hence not with one-way LOCC.

